Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,7,8)(13,14)(15,19)(16,18)(17,20);(1,2,4,3)(9,10,11,12)(13,15,18,17)(14,19,16,20);(2,3)(9,10)(11,12)(13,14,18,16)(15,20,17,19);(5,6)(7,8)(13,16,18,14)(15,19,17,20);(1,3,4,2)(13,17,18,15)(14,19,16,20)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=fg,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261066, #72261202, #72261204, … |
| maximal normal subgroup of | #72261066, #72261202, #72261204, … |
| maximal quotient of | #72332972 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2018454);